/*-
* SPDX-License-Identifier: BSD-2-Clause-FreeBSD
*
* Copyright (c) 2007-2013 Bruce D. Evans
* All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the above copyright
* notice unmodified, this list of conditions, and the following
* disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
*
* THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
* IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
* OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
* IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
* INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
* NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
* DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
* THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
* (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
* THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*/
#include <sys/cdefs.h>
__FBSDID("$FreeBSD$");
/**
* Implementation of the natural logarithm of x for Intel 80-bit format.
*
* First decompose x into its base 2 representation:
*
* log(x) = log(X * 2**k), where X is in [1, 2)
* = log(X) + k * log(2).
*
* Let X = X_i + e, where X_i is the center of one of the intervals
* [-1.0/256, 1.0/256), [1.0/256, 3.0/256), .... [2.0-1.0/256, 2.0+1.0/256)
* and X is in this interval. Then
*
* log(X) = log(X_i + e)
* = log(X_i * (1 + e / X_i))
* = log(X_i) + log(1 + e / X_i).
*
* The values log(X_i) are tabulated below. Let d = e / X_i and use
*
* log(1 + d) = p(d)
*
* where p(d) = d - 0.5*d*d + ... is a special minimax polynomial of
* suitably high degree.
*
* To get sufficiently small roundoff errors, k * log(2), log(X_i), and
* sometimes (if |k| is not large) the first term in p(d) must be evaluated
* and added up in extra precision. Extra precision is not needed for the
* rest of p(d). In the worst case when k = 0 and log(X_i) is 0, the final
* error is controlled mainly by the error in the second term in p(d). The
* error in this term itself is at most 0.5 ulps from the d*d operation in
* it. The error in this term relative to the first term is thus at most
* 0.5 * |-0.5| * |d| < 1.0/1024 ulps. We aim for an accumulated error of
* at most twice this at the point of the final rounding step. Thus the
* final error should be at most 0.5 + 1.0/512 = 0.5020 ulps. Exhaustive
* testing of a float variant of this function showed a maximum final error
* of 0.5008 ulps. Non-exhaustive testing of a double variant of this
* function showed a maximum final error of 0.5078 ulps (near 1+1.0/256).
*
* We made the maximum of |d| (and thus the total relative error and the
* degree of p(d)) small by using a large number of intervals. Using
* centers of intervals instead of endpoints reduces this maximum by a
* factor of 2 for a given number of intervals. p(d) is special only
* in beginning with the Taylor coefficients 0 + 1*d, which tends to happen
* naturally. The most accurate minimax polynomial of a given degree might
* be different, but then we wouldn't want it since we would have to do
* extra work to avoid roundoff error (especially for P0*d instead of d).
*/
#ifdef DEBUG
#include <assert.h>
#include <fenv.h>
#endif
#ifdef __i386__
#include <ieeefp.h>
#endif
#include "fpmath.h"
#include "math.h"
#define i386_SSE_GOOD
#ifndef NO_STRUCT_RETURN
#define STRUCT_RETURN
#endif
#include "math_private.h"
#if !defined(NO_UTAB) && !defined(NO_UTABL)
#define USE_UTAB
#endif
/*
* Domain [-0.005280, 0.004838], range ~[-5.1736e-22, 5.1738e-22]:
* |log(1 + d)/d - p(d)| < 2**-70.7
*/
static const double
P2 = -0.5,
P3 = 3.3333333333333359e-1, /* 0x1555555555555a.0p-54 */
P4 = -2.5000000000004424e-1, /* -0x1000000000031d.0p-54 */
P5 = 1.9999999992970016e-1, /* 0x1999999972f3c7.0p-55 */
P6 = -1.6666666072191585e-1, /* -0x15555548912c09.0p-55 */
P7 = 1.4286227413310518e-1, /* 0x12494f9d9def91.0p-55 */
P8 = -1.2518388626763144e-1; /* -0x1006068cc0b97c.0p-55 */
static volatile const double zero = 0;
#define INTERVALS 128
#define LOG2_INTERVALS 7
#define TSIZE (INTERVALS + 1)
#define G(i) (T[(i)].G)
#define F_hi(i) (T[(i)].F_hi)
#define F_lo(i) (T[(i)].F_lo)
#define ln2_hi F_hi(TSIZE - 1)
#define ln2_lo F_lo(TSIZE - 1)
#define E(i) (U[(i)].E)
#define H(i) (U[(i)].H)
static const struct {
float G; /* 1/(1 + i/128) rounded to 8/9 bits */
float F_hi; /* log(1 / G_i) rounded (see below) */
double F_lo; /* next 53 bits for log(1 / G_i) */
} T[TSIZE] = {
/*
* ln2_hi and each F_hi(i) are rounded to a number of bits that
* makes F_hi(i) + dk*ln2_hi exact for all i and all dk.
*
* The last entry (for X just below 2) is used to define ln2_hi
* and ln2_lo, to ensure that F_hi(i) and F_lo(i) cancel exactly
* with dk*ln2_hi and dk*ln2_lo, respectively, when dk = -1.
* This is needed for accuracy when x is just below 1. (To avoid
* special cases, such x are "reduced" strangely to X just below
* 2 and dk = -1, and then the exact cancellation is needed
* because any the error from any non-exactness would be too
* large).
*
* We want to share this table between double precision and ld80,
* so the relevant range of dk is the larger one of ld80
* ([-16445, 16383]) and the relevant exactness requirement is
* the stricter one of double precision. The maximum number of
* bits in F_hi(i) that works is very dependent on i but has
* a minimum of 33. We only need about 12 bits in F_hi(i) for
* it to provide enough extra precision in double precision (11
* more than that are required for ld80).
*
* We round F_hi(i) to 24 bits so that it can have type float,
* mainly to minimize the size of the table. Using all 24 bits
* in a float for it automatically satisfies the above constraints.
*/
{ 0x800000.0p-23, 0, 0 },
{ 0xfe0000.0p-24, 0x8080ac.0p-30, -0x14ee431dae6675.0p-84 },
{ 0xfc0000.0p-24, 0x8102b3.0p-29, -0x1db29ee2d83718.0p-84 },
{ 0xfa0000.0p-24, 0xc24929.0p-29, 0x1191957d173698.0p-83 },
{ 0xf80000.0p-24, 0x820aec.0p-28, 0x13ce8888e02e79.0p-82 },
{ 0xf60000.0p-24, 0xa33577.0p-28, -0x17a4382ce6eb7c.0p-82 },
{ 0xf48000.0p-24, 0xbc42cb.0p-28, -0x172a21161a1076.0p-83 },
{ 0xf30000.0p-24, 0xd57797.0p-28, -0x1e09de07cb9589.0p-82 },
{ 0xf10000.0p-24, 0xf7518e.0p-28, 0x1ae1eec1b036c5.0p-91 },
{ 0xef0000.0p-24, 0x8cb9df.0p-27, -0x1d7355325d560e.0p-81 },
{ 0xed8000.0p-24, 0x999ec0.0p-27, -0x1f9f02d256d503.0p-82 },
{ 0xec0000.0p-24, 0xa6988b.0p-27, -0x16fc0a9d12c17a.0p-83 },
{ 0xea0000.0p-24, 0xb80698.0p-27, 0x15d581c1e8da9a.0p-81 },
{ 0xe80000.0p-24, 0xc99af3.0p-27, -0x1535b3ba8f150b.0p-83 },
{ 0xe70000.0p-24, 0xd273b2.0p-27, 0x163786f5251af0.0p-85 },
{ 0xe50000.0p-24, 0xe442c0.0p-27, 0x1bc4b2368e32d5.0p-84 },
{ 0xe38000.0p-24, 0xf1b83f.0p-27, 0x1c6090f684e676.0p-81 },
{ 0xe20000.0p-24, 0xff448a.0p-27, -0x1890aa69ac9f42.0p-82 },
{ 0xe08000.0p-24, 0x8673f6.0p-26, 0x1b9985194b6b00.0p-80 },
{ 0xdf0000.0p-24, 0x8d515c.0p-26, -0x1dc08d61c6ef1e.0p-83 },
{ 0xdd8000.0p-24, 0x943a9e.0p-26, -0x1f72a2dac729b4.0p-82 },
{ 0xdc0000.0p-24, 0x9b2fe6.0p-26, -0x1fd4dfd3a0afb9.0p-80 },
{ 0xda8000.0p-24, 0xa2315d.0p-26, -0x11b26121629c47.0p-82 },
{ 0xd90000.0p-24, 0xa93f2f.0p-26, 0x1286d633e8e569.0p-81 },
{ 0xd78000.0p-24, 0xb05988.0p-26, 0x16128eba936770.0p-84 },
{ 0xd60000.0p-24, 0xb78094.0p-26, 0x16ead577390d32.0p-80 },
{ 0xd50000.0p-24, 0xbc4c6c.0p-26, 0x151131ccf7c7b7.0p-81 },
{ 0xd38000.0p-24, 0xc3890a.0p-26, -0x115e2cd714bd06.0p-80 },
{ 0xd20000.0p-24, 0xcad2d7.0p-26, -0x1847f406ebd3b0.0p-82 },
{ 0xd10000.0p-24, 0xcfb620.0p-26, 0x1c2259904d6866.0p-81 },
{ 0xcf8000.0p-24, 0xd71653.0p-26, 0x1ece57a8d5ae55.0p-80 },
{ 0xce0000.0p-24, 0xde843a.0p-26, -0x1f109d4bc45954.0p-81 },
{ 0xcd0000.0p-24, 0xe37fde.0p-26, 0x1bc03dc271a74d.0p-81 },
{ 0xcb8000.0p-24, 0xeb050c.0p-26, -0x1bf2badc0df842.0p-85 },
{ 0xca0000.0p-24, 0xf29878.0p-26, -0x18efededd89fbe.0p-87 },
{ 0xc90000.0p-24, 0xf7ad6f.0p-26, 0x1373ff977baa69.0p-81 },
{ 0xc80000.0p-24, 0xfcc8e3.0p-26, 0x196766f2fb3283.0p-80 },
{ 0xc68000.0p-24, 0x823f30.0p-25, 0x19bd076f7c434e.0p-79 },
{ 0xc58000.0p-24, 0x84d52c.0p-25, -0x1a327257af0f46.0p-79 },
{ 0xc40000.0p-24, 0x88bc74.0p-25, 0x113f23def19c5a.0p-81 },
{ 0xc30000.0p-24, 0x8b5ae6.0p-25, 0x1759f6e6b37de9.0p-79 },
{ 0xc20000.0p-24, 0x8dfccb.0p-25, 0x1ad35ca6ed5148.0p-81 },
{ 0xc10000.0p-24, 0x90a22b.0p-25, 0x1a1d71a87deba4.0p-79 },
{ 0xbf8000.0p-24, 0x94a0d8.0p-25, -0x139e5210c2b731.0p-80 },
{ 0xbe8000.0p-24, 0x974f16.0p-25, -0x18f6ebcff3ed73.0p-81 },
{ 0xbd8000.0p-24, 0x9a00f1.0p-25, -0x1aa268be39aab7.0p-79 },
{ 0xbc8000.0p-24, 0x9cb672.0p-25, -0x14c8815839c566.0p-79 },
{ 0xbb0000.0p-24, 0xa0cda1.0p-25, 0x1eaf46390dbb24.0p-81 },
{ 0xba0000.0p-24, 0xa38c6e.0p-25, 0x138e20d831f698.0p-81 },
{ 0xb90000.0p-24, 0xa64f05.0p-25, -0x1e8d3c41123616.0p-82 },
{ 0xb80000.0p-24, 0xa91570.0p-25, 0x1ce28f5f3840b2.0p-80 },
{ 0xb70000.0p-24, 0xabdfbb.0p-25, -0x186e5c0a424234.0p-79 },
{ 0xb60000.0p-24, 0xaeadef.0p-25, -0x14d41a0b2a08a4.0p-83 },
{ 0xb50000.0p-24, 0xb18018.0p-25, 0x16755892770634.0p-79 },
{ 0xb40000.0p-24, 0xb45642.0p-25, -0x16395ebe59b152.0p-82 },
{ 0xb30000.0p-24, 0xb73077.0p-25, 0x1abc65c8595f09.0p-80 },
{ 0xb20000.0p-24, 0xba0ec4.0p-25, -0x1273089d3dad89.0p-79 },
{ 0xb10000.0p-24, 0xbcf133.0p-25, 0x10f9f67b1f4bbf.0p-79 },
{ 0xb00000.0p-24, 0xbfd7d2.0p-25, -0x109fab90486409.0p-80 },
{ 0xaf0000.0p-24, 0xc2c2ac.0p-25, -0x1124680aa43333.0p-79 },
{ 0xae8000.0p-24, 0xc439b3.0p-25, -0x1f360cc4710fc0.0p-80 },
{ 0xad8000.0p-24, 0xc72afd.0p-25, -0x132d91f21d89c9.0p-80 },
{ 0xac8000.0p-24, 0xca20a2.0p-25, -0x16bf9b4d1f8da8.0p-79 },
{ 0xab8000.0p-24, 0xcd1aae.0p-25, 0x19deb5ce6a6a87.0p-81 },
{ 0xaa8000.0p-24, 0xd0192f.0p-25, 0x1a29fb48f7d3cb.0p-79 },
{ 0xaa0000.0p-24, 0xd19a20.0p-25, 0x1127d3c6457f9d.0p-81 },
{ 0xa90000.0p-24, 0xd49f6a.0p-25, -0x1ba930e486a0ac.0p-81 },
{ 0xa80000.0p-24, 0xd7a94b.0p-25, -0x1b6e645f31549e.0p-79 },
{ 0xa70000.0p-24, 0xdab7d0.0p-25, 0x1118a425494b61.0p-80 },
{ 0xa68000.0p-24, 0xdc40d5.0p-25, 0x1966f24d29d3a3.0p-80 },
{ 0xa58000.0p-24, 0xdf566d.0p-25, -0x1d8e52eb2248f1.0p-82 },
{ 0xa48000.0p-24, 0xe270ce.0p-25, -0x1ee370f96e6b68.0p-80 },
{ 0xa40000.0p-24, 0xe3ffce.0p-25, 0x1d155324911f57.0p-80 },
{ 0xa30000.0p-24, 0xe72179.0p-25, -0x1fe6e2f2f867d9.0p-80 },
{ 0xa20000.0p-24, 0xea4812.0p-25, 0x1b7be9add7f4d4.0p-80 },
{ 0xa18000.0p-24, 0xebdd3d.0p-25, 0x1b3cfb3f7511dd.0p-79 },
{ 0xa08000.0p-24, 0xef0b5b.0p-25, -0x1220de1f730190.0p-79 },
{ 0xa00000.0p-24, 0xf0a451.0p-25, -0x176364c9ac81cd.0p-80 },
{ 0x9f0000.0p-24, 0xf3da16.0p-25, 0x1eed6b9aafac8d.0p-81 },
{ 0x9e8000.0p-24, 0xf576e9.0p-25, 0x1d593218675af2.0p-79 },
{ 0x9d8000.0p-24, 0xf8b47c.0p-25, -0x13e8eb7da053e0.0p-84 },
{ 0x9d0000.0p-24, 0xfa553f.0p-25, 0x1c063259bcade0.0p-79 },
{ 0x9c0000.0p-24, 0xfd9ac5.0p-25, 0x1ef491085fa3c1.0p-79 },
{ 0x9b8000.0p-24, 0xff3f8c.0p-25, 0x1d607a7c2b8c53.0p-79 },
{ 0x9a8000.0p-24, 0x814697.0p-24, -0x12ad3817004f3f.0p-78 },
{ 0x9a0000.0p-24, 0x821b06.0p-24, -0x189fc53117f9e5.0p-81 },
{ 0x990000.0p-24, 0x83c5f8.0p-24, 0x14cf15a048907b.0p-79 },
{ 0x988000.0p-24, 0x849c7d.0p-24, 0x1cbb1d35fb8287.0p-78 },
{ 0x978000.0p-24, 0x864ba6.0p-24, 0x1128639b814f9c.0p-78 },
{ 0x970000.0p-24, 0x87244c.0p-24, 0x184733853300f0.0p-79 },
{ 0x968000.0p-24, 0x87fdaa.0p-24, 0x109d23aef77dd6.0p-80 },
{ 0x958000.0p-24, 0x89b293.0p-24, -0x1a81ef367a59de.0p-78 },
{ 0x950000.0p-24, 0x8a8e20.0p-24, -0x121ad3dbb2f452.0p-78 },
{ 0x948000.0p-24, 0x8b6a6a.0p-24, -0x1cfb981628af72.0p-79 },
{ 0x938000.0p-24, 0x8d253a.0p-24, -0x1d21730ea76cfe.0p-79 },
{ 0x930000.0p-24, 0x8e03c2.0p-24, 0x135cc00e566f77.0p-78 },
{ 0x928000.0p-24, 0x8ee30d.0p-24, -0x10fcb5df257a26.0p-80 },
{ 0x918000.0p-24, 0x90a3ee.0p-24, -0x16e171b15433d7.0p-79 },
{ 0x910000.0p-24, 0x918587.0p-24, -0x1d050da07f3237.0p-79 },
{ 0x908000.0p-24, 0x9267e7.0p-24, 0x1be03669a5268d.0p-79 },
{ 0x8f8000.0p-24, 0x942f04.0p-24, 0x10b28e0e26c337.0p-79 },
{ 0x8f0000.0p-24, 0x9513c3.0p-24, 0x1a1d820da57cf3.0p-78 },
{ 0x8e8000.0p-24, 0x95f950.0p-24, -0x19ef8f13ae3cf1.0p-79 },
{ 0x8e0000.0p-24, 0x96dfab.0p-24, -0x109e417a6e507c.0p-78 },
{ 0x8d0000.0p-24, 0x98aed2.0p-24, 0x10d01a2c5b0e98.0p-79 },
{ 0x8c8000.0p-24, 0x9997a2.0p-24, -0x1d6a50d4b61ea7.0p-78 },
{ 0x8c0000.0p-24, 0x9a8145.0p-24, 0x1b3b190b83f952.0p-78 },
{ 0x8b8000.0p-24, 0x9b6bbf.0p-24, 0x13a69fad7e7abe.0p-78 },
{ 0x8b0000.0p-24, 0x9c5711.0p-24, -0x11cd12316f576b.0p-78 },
{ 0x8a8000.0p-24, 0x9d433b.0p-24, 0x1c95c444b807a2.0p-79 },
{ 0x898000.0p-24, 0x9f1e22.0p-24, -0x1b9c224ea698c3.0p-79 },
{ 0x890000.0p-24, 0xa00ce1.0p-24, 0x125ca93186cf0f.0p-81 },
{ 0x888000.0p-24, 0xa0fc80.0p-24, -0x1ee38a7bc228b3.0p-79 },
{ 0x880000.0p-24, 0xa1ed00.0p-24, -0x1a0db876613d20.0p-78 },
{ 0x878000.0p-24, 0xa2de62.0p-24, 0x193224e8516c01.0p-79 },
{ 0x870000.0p-24, 0xa3d0a9.0p-24, 0x1fa28b4d2541ad.0p-79 },
{ 0x868000.0p-24, 0xa4c3d6.0p-24, 0x1c1b5760fb4572.0p-78 },
{ 0x858000.0p-24, 0xa6acea.0p-24, 0x1fed5d0f65949c.0p-80 },
{ 0x850000.0p-24, 0xa7a2d4.0p-24, 0x1ad270c9d74936.0p-80 },
{ 0x848000.0p-24, 0xa899ab.0p-24, 0x199ff15ce53266.0p-79 },
{ 0x840000.0p-24, 0xa99171.0p-24, 0x1a19e15ccc45d2.0p-79 },
{ 0x838000.0p-24, 0xaa8a28.0p-24, -0x121a14ec532b36.0p-80 },
{ 0x830000.0p-24, 0xab83d1.0p-24, 0x1aee319980bff3.0p-79 },
{ 0x828000.0p-24, 0xac7e6f.0p-24, -0x18ffd9e3900346.0p-80 },
{ 0x820000.0p-24, 0xad7a03.0p-24, -0x1e4db102ce29f8.0p-80 },
{ 0x818000.0p-24, 0xae768f.0p-24, 0x17c35c55a04a83.0p-81 },
{ 0x810000.0p-24, 0xaf7415.0p-24, 0x1448324047019b.0p-78 },
{ 0x808000.0p-24, 0xb07298.0p-24, -0x1750ee3915a198.0p-78 },
{ 0x800000.0p-24, 0xb17218.0p-24, -0x105c610ca86c39.0p-81 },
};
#ifdef USE_UTAB
static const struct {
float H; /* 1 + i/INTERVALS (exact) */
float E; /* H(i) * G(i) - 1 (exact) */
} U[TSIZE] = {
{ 0x800000.0p-23, 0 },
{ 0x810000.0p-23, -0x800000.0p-37 },
{ 0x820000.0p-23, -0x800000.0p-35 },
{ 0x830000.0p-23, -0x900000.0p-34 },
{ 0x840000.0p-23, -0x800000.0p-33 },
{ 0x850000.0p-23, -0xc80000.0p-33 },
{ 0x860000.0p-23, -0xa00000.0p-36 },
{ 0x870000.0p-23, 0x940000.0p-33 },
{ 0x880000.0p-23, 0x800000.0p-35 },
{ 0x890000.0p-23, -0xc80000.0p-34 },
{ 0x8a0000.0p-23, 0xe00000.0p-36 },
{ 0x8b0000.0p-23, 0x900000.0p-33 },
{ 0x8c0000.0p-23, -0x800000.0p-35 },
{ 0x8d0000.0p-23, -0xe00000.0p-33 },
{ 0x8e0000.0p-23, 0x880000.0p-33 },
{ 0x8f0000.0p-23, -0xa80000.0p-34 },
{ 0x900000.0p-23, -0x800000.0p-35 },
{ 0x910000.0p-23, 0x800000.0p-37 },
{ 0x920000.0p-23, 0x900000.0p-35 },
{ 0x930000.0p-23, 0xd00000.0p-35 },
{ 0x940000.0p-23, 0xe00000.0p-35 },
{ 0x950000.0p-23, 0xc00000.0p-35 },
{ 0x960000.0p-23, 0xe00000.0p-36 },
{ 0x970000.0p-23, -0x800000.0p-38 },
{ 0x980000.0p-23, -0xc00000.0p-35 },
{ 0x990000.0p-23, -0xd00000.0p-34 },
{ 0x9a0000.0p-23, 0x880000.0p-33 },
{ 0x9b0000.0p-23, 0xe80000.0p-35 },
{ 0x9c0000.0p-23, -0x800000.0p-35 },
{ 0x9d0000.0p-23, 0xb40000.0p-33 },
{ 0x9e0000.0p-23, 0x880000.0p-34 },
{ 0x9f0000.0p-23, -0xe00000.0p-35 },
{ 0xa00000.0p-23, 0x800000.0p-33 },
{ 0xa10000.0p-23, -0x900000.0p-36 },
{ 0xa20000.0p-23, -0xb00000.0p-33 },
{ 0xa30000.0p-23, -0xa00000.0p-36 },
{ 0xa40000.0p-23, 0x800000.0p-33 },
{ 0xa50000.0p-23, -0xf80000.0p-35 },
{ 0xa60000.0p-23, 0x880000.0p-34 },
{ 0xa70000.0p-23, -0x900000.0p-33 },
{ 0xa80000.0p-23, -0x800000.0p-35 },
{ 0xa90000.0p-23, 0x900000.0p-34 },
{ 0xaa0000.0p-23, 0xa80000.0p-33 },
{ 0xab0000.0p-23, -0xac0000.0p-34 },
{ 0xac0000.0p-23, -0x800000.0p-37 },
{ 0xad0000.0p-23, 0xf80000.0p-35 },
{ 0xae0000.0p-23, 0xf80000.0p-34 },
{ 0xaf0000.0p-23, -0xac0000.0p-33 },
{ 0xb00000.0p-23, -0x800000.0p-33 },
{ 0xb10000.0p-23, -0xb80000.0p-34 },
{ 0xb20000.0p-23, -0x800000.0p-34 },
{ 0xb30000.0p-23, -0xb00000.0p-35 },
{ 0xb40000.0p-23, -0x800000.0p-35 },
{ 0xb50000.0p-23, -0xe00000.0p-36 },
{ 0xb60000.0p-23, -0x800000.0p-35 },
{ 0xb70000.0p-23, -0xb00000.0p-35 },
{ 0xb80000.0p-23, -0x800000.0p-34 },
{ 0xb90000.0p-23, -0xb80000.0p-34 },
{ 0xba0000.0p-23, -0x800000.0p-33 },
{ 0xbb0000.0p-23, -0xac0000.0p-33 },
{ 0xbc0000.0p-23, 0x980000.0p-33 },
{ 0xbd0000.0p-23, 0xbc0000.0p-34 },
{ 0xbe0000.0p-23, 0xe00000.0p-36 },
{ 0xbf0000.0p-23, -0xb80000.0p-35 },
{ 0xc00000.0p-23, -0x800000.0p-33 },
{ 0xc10000.0p-23, 0xa80000.0p-33 },
{ 0xc20000.0p-23, 0x900000.0p-34 },
{ 0xc30000.0p-23, -0x800000.0p-35 },
{ 0xc40000.0p-23, -0x900000.0p-33 },
{ 0xc50000.0p-23, 0x820000.0p-33 },
{ 0xc60000.0p-23, 0x800000.0p-38 },
{ 0xc70000.0p-23, -0x820000.0p-33 },
{ 0xc80000.0p-23, 0x800000.0p-33 },
{ 0xc90000.0p-23, -0xa00000.0p-36 },
{ 0xca0000.0p-23, -0xb00000.0p-33 },
{ 0xcb0000.0p-23, 0x840000.0p-34 },
{ 0xcc0000.0p-23, -0xd00000.0p-34 },
{ 0xcd0000.0p-23, 0x800000.0p-33 },
{ 0xce0000.0p-23, -0xe00000.0p-35 },
{ 0xcf0000.0p-23, 0xa60000.0p-33 },
{ 0xd00000.0p-23, -0x800000.0p-35 },
{ 0xd10000.0p-23, 0xb40000.0p-33 },
{ 0xd20000.0p-23, -0x800000.0p-35 },
{ 0xd30000.0p-23, 0xaa0000.0p-33 },
{ 0xd40000.0p-23, -0xe00000.0p-35 },
{ 0xd50000.0p-23, 0x880000.0p-33 },
{ 0xd60000.0p-23, -0xd00000.0p-34 },
{ 0xd70000.0p-23, 0x9c0000.0p-34 },
{ 0xd80000.0p-23, -0xb00000.0p-33 },
{ 0xd90000.0p-23, -0x800000.0p-38 },
{ 0xda0000.0p-23, 0xa40000.0p-33 },
{ 0xdb0000.0p-23, -0xdc0000.0p-34 },
{ 0xdc0000.0p-23, 0xc00000.0p-35 },
{ 0xdd0000.0p-23, 0xca0000.0p-33 },
{ 0xde0000.0p-23, -0xb80000.0p-34 },
{ 0xdf0000.0p-23, 0xd00000.0p-35 },
{ 0xe00000.0p-23, 0xc00000.0p-33 },
{ 0xe10000.0p-23, -0xf40000.0p-34 },
{ 0xe20000.0p-23, 0x800000.0p-37 },
{ 0xe30000.0p-23, 0x860000.0p-33 },
{ 0xe40000.0p-23, -0xc80000.0p-33 },
{ 0xe50000.0p-23, -0xa80000.0p-34 },
{ 0xe60000.0p-23, 0xe00000.0p-36 },
{ 0xe70000.0p-23, 0x880000.0p-33 },
{ 0xe80000.0p-23, -0xe00000.0p-33 },
{ 0xe90000.0p-23, -0xfc0000.0p-34 },
{ 0xea0000.0p-23, -0x800000.0p-35 },
{ 0xeb0000.0p-23, 0xe80000.0p-35 },
{ 0xec0000.0p-23, 0x900000.0p-33 },
{ 0xed0000.0p-23, 0xe20000.0p-33 },
{ 0xee0000.0p-23, -0xac0000.0p-33 },
{ 0xef0000.0p-23, -0xc80000.0p-34 },
{ 0xf00000.0p-23, -0x800000.0p-35 },
{ 0xf10000.0p-23, 0x800000.0p-35 },
{ 0xf20000.0p-23, 0xb80000.0p-34 },
{ 0xf30000.0p-23, 0x940000.0p-33 },
{ 0xf40000.0p-23, 0xc80000.0p-33 },
{ 0xf50000.0p-23, -0xf20000.0p-33 },
{ 0xf60000.0p-23, -0xc80000.0p-33 },
{ 0xf70000.0p-23, -0xa20000.0p-33 },
{ 0xf80000.0p-23, -0x800000.0p-33 },
{ 0xf90000.0p-23, -0xc40000.0p-34 },
{ 0xfa0000.0p-23, -0x900000.0p-34 },
{ 0xfb0000.0p-23, -0xc80000.0p-35 },
{ 0xfc0000.0p-23, -0x800000.0p-35 },
{ 0xfd0000.0p-23, -0x900000.0p-36 },
{ 0xfe0000.0p-23, -0x800000.0p-37 },
{ 0xff0000.0p-23, -0x800000.0p-39 },
{ 0x800000.0p-22, 0 },
};
#endif /* USE_UTAB */
#ifdef STRUCT_RETURN
#define RETURN1(rp, v) do { \
(rp)->hi = (v); \
(rp)->lo_set = 0; \
return; \
} while (0)
#define RETURN2(rp, h, l) do { \
(rp)->hi = (h); \
(rp)->lo = (l); \
(rp)->lo_set = 1; \
return; \
} while (0)
struct ld {
long double hi;
long double lo;
int lo_set;
};
#else
#define RETURN1(rp, v) RETURNF(v)
#define RETURN2(rp, h, l) RETURNI((h) + (l))
#endif
#ifdef STRUCT_RETURN
static inline __always_inline void
k_logl(long double x, struct ld *rp)
#else
long double
logl(long double x)
#endif
{
long double d, dk, val_hi, val_lo, z;
uint64_t ix, lx;
int i, k;
uint16_t hx;
EXTRACT_LDBL80_WORDS(hx, lx, x);
k = -16383;
#if 0 /* Hard to do efficiently. Don't do it until we support all modes. */
if (x == 1)
RETURN1(rp, 0); /* log(1) = +0 in all rounding modes */
#endif
if (hx == 0 || hx >= 0x8000) { /* zero, negative or subnormal? */
if (((hx & 0x7fff) | lx) == 0)
RETURN1(rp, -1 / zero); /* log(+-0) = -Inf */
if (hx != 0)
/* log(neg or [pseudo-]NaN) = qNaN: */
RETURN1(rp, (x - x) / zero);
x *= 0x1.0p65; /* subnormal; scale up x */
/* including pseudo-subnormals */
EXTRACT_LDBL80_WORDS(hx, lx, x);
k = -16383 - 65;
} else if (hx >= 0x7fff || (lx & 0x8000000000000000ULL) == 0)
RETURN1(rp, x + x); /* log(Inf or NaN) = Inf or qNaN */
/* log(pseudo-Inf) = qNaN */
/* log(pseudo-NaN) = qNaN */
/* log(unnormal) = qNaN */
#ifndef STRUCT_RETURN
ENTERI();
#endif
k += hx;
ix = lx & 0x7fffffffffffffffULL;
dk = k;
/* Scale x to be in [1, 2). */
SET_LDBL_EXPSIGN(x, 0x3fff);
/* 0 <= i <= INTERVALS: */
#define L2I (64 - LOG2_INTERVALS)
i = (ix + (1LL << (L2I - 2))) >> (L2I - 1);
/*
* -0.005280 < d < 0.004838. In particular, the infinite-
* precision |d| is <= 2**-7. Rounding of G(i) to 8 bits
* ensures that d is representable without extra precision for
* this bound on |d| (since when this calculation is expressed
* as x*G(i)-1, the multiplication needs as many extra bits as
* G(i) has and the subtraction cancels 8 bits). But for
* most i (107 cases out of 129), the infinite-precision |d|
* is <= 2**-8. G(i) is rounded to 9 bits for such i to give
* better accuracy (this works by improving the bound on |d|,
* which in turn allows rounding to 9 bits in more cases).
* This is only important when the original x is near 1 -- it
* lets us avoid using a special method to give the desired
* accuracy for such x.
*/
if (0)
d = x * G(i) - 1;
else {
#ifdef USE_UTAB
d = (x - H(i)) * G(i) + E(i);
#else
long double x_hi, x_lo;
float fx_hi;
/*
* Split x into x_hi + x_lo to calculate x*G(i)-1 exactly.
* G(i) has at most 9 bits, so the splitting point is not
* critical.
*/
SET_FLOAT_WORD(fx_hi, (lx >> 40) | 0x3f800000);
x_hi = fx_hi;
x_lo = x - x_hi;
d = x_hi * G(i) - 1 + x_lo * G(i);
#endif
}
/*
* Our algorithm depends on exact cancellation of F_lo(i) and
* F_hi(i) with dk*ln_2_lo and dk*ln2_hi when k is -1 and i is
* at the end of the table. This and other technical complications
* make it difficult to avoid the double scaling in (dk*ln2) *
* log(base) for base != e without losing more accuracy and/or
* efficiency than is gained.
*/
z = d * d;
val_lo = z * d * z * (z * (d * P8 + P7) + (d * P6 + P5)) +
(F_lo(i) + dk * ln2_lo + z * d * (d * P4 + P3)) + z * P2;
val_hi = d;
#ifdef DEBUG
if (fetestexcept(FE_UNDERFLOW))
breakpoint();
#endif
_3sumF(val_hi, val_lo, F_hi(i) + dk * ln2_hi);
RETURN2(rp, val_hi, val_lo);
}
long double
log1pl(long double x)
{
long double d, d_hi, d_lo, dk, f_lo, val_hi, val_lo, z;
long double f_hi, twopminusk;
uint64_t ix, lx;
int i, k;
int16_t ax, hx;
DOPRINT_START(&x);
EXTRACT_LDBL80_WORDS(hx, lx, x);
if (hx < 0x3fff) { /* x < 1, or x neg NaN */
ax = hx & 0x7fff;
if (ax >= 0x3fff) { /* x <= -1, or x neg NaN */
if (ax == 0x3fff && lx == 0x8000000000000000ULL)
RETURNP(-1 / zero); /* log1p(-1) = -Inf */
/* log1p(x < 1, or x [pseudo-]NaN) = qNaN: */
RETURNP((x - x) / (x - x));
}
if (ax <= 0x3fbe) { /* |x| < 2**-64 */
if ((int)x == 0)
RETURNP(x); /* x with inexact if x != 0 */
}
f_hi = 1;
f_lo = x;
} else if (hx >= 0x7fff) { /* x +Inf or non-neg NaN */
RETURNP(x + x); /* log1p(Inf or NaN) = Inf or qNaN */
/* log1p(pseudo-Inf) = qNaN */
/* log1p(pseudo-NaN) = qNaN */
/* log1p(unnormal) = qNaN */
} else if (hx < 0x407f) { /* 1 <= x < 2**128 */
f_hi = x;
f_lo = 1;
} else { /* 2**128 <= x < +Inf */
f_hi = x;
f_lo = 0; /* avoid underflow of the P5 term */
}
ENTERI();
x = f_hi + f_lo;
f_lo = (f_hi - x) + f_lo;
EXTRACT_LDBL80_WORDS(hx, lx, x);
k = -16383;
k += hx;
ix = lx & 0x7fffffffffffffffULL;
dk = k;
SET_LDBL_EXPSIGN(x, 0x3fff);
twopminusk = 1;
SET_LDBL_EXPSIGN(twopminusk, 0x7ffe - (hx & 0x7fff));
f_lo *= twopminusk;
i = (ix + (1LL << (L2I - 2))) >> (L2I - 1);
/*
* x*G(i)-1 (with a reduced x) can be represented exactly, as
* above, but now we need to evaluate the polynomial on d =
* (x+f_lo)*G(i)-1 and extra precision is needed for that.
* Since x+x_lo is a hi+lo decomposition and subtracting 1
* doesn't lose too many bits, an inexact calculation for
* f_lo*G(i) is good enough.
*/
if (0)
d_hi = x * G(i) - 1;
else {
#ifdef USE_UTAB
d_hi = (x - H(i)) * G(i) + E(i);
#else
long double x_hi, x_lo;
float fx_hi;
SET_FLOAT_WORD(fx_hi, (lx >> 40) | 0x3f800000);
x_hi = fx_hi;
x_lo = x - x_hi;
d_hi = x_hi * G(i) - 1 + x_lo * G(i);
#endif
}
d_lo = f_lo * G(i);
/*
* This is _2sumF(d_hi, d_lo) inlined. The condition
* (d_hi == 0 || |d_hi| >= |d_lo|) for using _2sumF() is not
* always satisifed, so it is not clear that this works, but
* it works in practice. It works even if it gives a wrong
* normalized d_lo, since |d_lo| > |d_hi| implies that i is
* nonzero and d is tiny, so the F(i) term dominates d_lo.
* In float precision:
* (By exhaustive testing, the worst case is d_hi = 0x1.bp-25.
* And if d is only a little tinier than that, we would have
* another underflow problem for the P3 term; this is also ruled
* out by exhaustive testing.)
*/
d = d_hi + d_lo;
d_lo = d_hi - d + d_lo;
d_hi = d;
z = d * d;
val_lo = z * d * z * (z * (d * P8 + P7) + (d * P6 + P5)) +
(F_lo(i) + dk * ln2_lo + d_lo + z * d * (d * P4 + P3)) + z * P2;
val_hi = d_hi;
#ifdef DEBUG
if (fetestexcept(FE_UNDERFLOW))
breakpoint();
#endif
_3sumF(val_hi, val_lo, F_hi(i) + dk * ln2_hi);
RETURN2PI(val_hi, val_lo);
}
#ifdef STRUCT_RETURN
long double
logl(long double x)
{
struct ld r;
ENTERI();
DOPRINT_START(&x);
k_logl(x, &r);
RETURNSPI(&r);
}
/* Use macros since GCC < 8 rejects static const expressions in initializers. */
#define invln10_hi 4.3429448190317999e-1 /* 0x1bcb7b1526e000.0p-54 */
#define invln10_lo 7.1842412889749798e-14 /* 0x1438ca9aadd558.0p-96 */
#define invln2_hi 1.4426950408887933e0 /* 0x171547652b8000.0p-52 */
#define invln2_lo 1.7010652264631490e-13 /* 0x17f0bbbe87fed0.0p-95 */
/* Let the compiler pre-calculate this sum to avoid FE_INEXACT at run time. */
static const double invln10_lo_plus_hi = invln10_lo + invln10_hi;
static const double invln2_lo_plus_hi = invln2_lo + invln2_hi;
long double
log10l(long double x)
{
struct ld r;
long double hi, lo;
ENTERI();
DOPRINT_START(&x);
k_logl(x, &r);
if (!r.lo_set)
RETURNPI(r.hi);
_2sumF(r.hi, r.lo);
hi = (float)r.hi;
lo = r.lo + (r.hi - hi);
RETURN2PI(invln10_hi * hi,
invln10_lo_plus_hi * lo + invln10_lo * hi);
}
long double
log2l(long double x)
{
struct ld r;
long double hi, lo;
ENTERI();
DOPRINT_START(&x);
k_logl(x, &r);
if (!r.lo_set)
RETURNPI(r.hi);
_2sumF(r.hi, r.lo);
hi = (float)r.hi;
lo = r.lo + (r.hi - hi);
RETURN2PI(invln2_hi * hi,
invln2_lo_plus_hi * lo + invln2_lo * hi);
}
#endif /* STRUCT_RETURN */
